In physics, the integral of the Berry connection \(\oint A_{i}\,{\rm d}x_{i}\)  is the phase shift a quantum mechanical wave function obtains after being adiabatically moved through a parameter space with coordinates \(x_i\)  [1]}. Our surface tangent `state vector' \(|\psi \rangle \) , undergoes the same phase shift as we encircle a critical point. We can see this by examining how the vector changes as we move a small distance on the surface
\(\langle \psi |{\rm d}|\psi \rangle =\langle \psi |\frac{1}{\sqrt{2}}\left(-{e}_{2}+{\rm i}{e}_{1}\right){\rm d}\theta =\left(1+{\rm i}{\rm d}\theta \right)={\rm e}^{{\rm i}{\rm d}\theta }.\) 
